The induced subgraph K_2_,_3 in a non-Hamiltonian graphs

TL;DR

This paper characterizes non-Hamiltonian graphs by showing that a norm graph is non-Hamiltonian if and only if it contains a subgraph homeomorphic to K_{2,3}, linking structural properties to Hamiltonicity.

Contribution

It provides a new characterization of non-Hamiltonian graphs based on the presence of a specific subgraph, K_{2,3}, within reduced norm graphs.

Findings

Norm graphs are non-Hamiltonian iff they contain a K_{2,3} subgraph

Homeomorphism to K_{2,3} characterizes non-Hamiltonian norm graphs

Structural graph properties determine Hamiltonicity in this class

Abstract

A graph $\textit{G}$ is a tuple $(\textit{V}, \textit{E})$, where $\textit{V}$ is the vertex set, $\textit{E}$ is the edge set. A reduced graph is a graph of deleting non-Hamiltonian edges and smoothing out the redundant vertices of degree 2 on an edge except for leaving only one vertex of degree 2. We denote by I a set of cycles only jointed by inside vertices. |I| is the number of sets I in a graph. We use a norm graph to denote a reduced graph of |I|=1. $\textit{g}$ is a subgraph obtained by deleting all removable cycles from a basis of a norm graph. In this paper, we show that a norm graph $\textit{G}$ is non-Hamiltonian, if and only if, $\textit{g}$ and K_2_,_3 are homeomorphic.